In this paper, we investigate interacting topological properties of spinless and spin-1/2 SSH4 models in 1D optical superlattices, based on a state-of-the-art
density-matrix renormalization-group (DMRG) numerical method [1]}, [2]}.
For interacting SSH4 model, the topological invariant and classification of interacting TI become \(\mathbb {Z}_{4}\) , which are different from the single-particle TI classified with \(\mathbb {Z}\)  group.
The nearest-neighbor interaction can drive the topological insulator (TI) and the topologically trivial insulator phases to the charge density wave (CDW) phase, which is characterized with the entanglement spectrum, entanglement entropies, energy gaps, and CDW order parameter. We numerically work out the many-body phase diagrams, and show the typical features of the appeared quantum phases. Although the phase diagram is similar to that of interacting SSH model, we find the phase transitions to the CDW phase are continuous phase transitions, unlike those in the SSH model, which are argued to be first-order phase transitions based on variational study [3]}. The central charges at the phase boundaries between the CDW and TI/trivial insulator (TI and trivial insulator) phase are shown to be 2 (1). Further, we analyze the ground states of interacting spinful SSH4 model.
It shows that the repulsive on-site interaction of spin-1/2 SSH4 model can enhance the TI phase and suppress the
CDW phase, but the attractive on-site interaction plays the opposite role. Our results may stimulate a new avenue for simulating interacting fermionic topological phases using cold atom in optical lattices.
